where \(\tau _r(k)\)  is the scale-dependent relaxation time which can be
identified with the correlation time \(\tau (k)\)  of the
turbulent velocity field for large fluid Reynolds
numbers and large Péclet numbers.
Here functions with superscript \((0)\)  correspond to background
turbulence with zero turbulent heat flux.
Therefore, equation (REF ) is reduced to
\(\hat{\cal M} \Theta ^{\rm (III)} = - \left\langle \theta ({k}_1)\, \theta ({k}_2) \right\rangle / \tau (k)\) .
Validation of the \(\tau \)  approximation for different
situations has been performed in various numerical
simulations [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}.
We apply the \(\tau \)  approximation only to
study the deviations from the background turbulence which are
caused by the spatial derivatives of the mean temperature. The
background compressible density stratified turbulence
is assumed to be known (see below).
